Lectures a brief history of mine
   Inflation: An Open and Shut Case (April '98)
   Slides and audio for this talk can be downloaded from the ITP at UCSB.
   This talk will be based on joint work with Neil Turok, at Cambridge.
   Neil has always been interested in what might be called, alternative
   cosmology. He pushed the idea that topological defects like cosmic
   strings or textures, were the origin of the large scale structure of
   the universe. And he was a proponent of what is called, open
   inflation. This is the idea that the universe is infinitely large, and
   of low density, despite having been through a period of exponential
   expansion, in the very early stages. My opinion was, that these were
   all nice ideas, but that nature probably hadn't chosen the use any of
   them. I included open inflation in that list, because I believed
   strongly that the universe came into being, at a finite size, and I
   felt that implied that the universe now, was still of finite size, or
   closed. However, after Neil gave a seminar on open inflation in
   Cambridge, we got talking. We realized it was possible for the
   universe to come into existence, at a finite size, but nevertheless,
   be either a finite, or an infinitely large universe now. My talk will
   be about this idea, and new developments that have occur since then.
   One of these, which is so recent that it is not yet fully worked out,
   is that it seems there is an observational signature, of the kind of
   inflation Neil and I are proposing. The present measurements are not
   sensitive enough to see this effect, but it should be possible to test
   for it, in the observations the Planck satellite will make.
   Another recent development, is that observations of supernovas, have
   suggested that the universe may have a small cosmological constant, at
   the present time. Even before these observations, Neil and I had
   realized that if there was a four form gauge field, one could invoke
   anthropic arguments to make it cancel the cosmological constant, that
   one would expect from symmetry breaking. But the anthropic argument
   would not require it to cancel exactly. So there could be a small
   residual cosmological constant. This is exciting. I shall describe the
   observational evidence later.
   As you probably know, the universe is remarkably isotropic on a large
   scale. That is to say, it looks the same in all directions, if one
   goes beyond such local irregularities as the Milky Way, and the Local
   Group of galaxies. By far the most accurate measurement of the
   isotropy of the universe, is the faint background of microwave
   radiation, first discovered in 1965. At the present time at least, the
   universe is transparant to microwaves, in directions out of the plane
   of our galaxy. Thus the microwave background, must have propagated to
   us, from distances of the order of the Hubble radius, or greater. It
   should therefore give a sensitive measurement, of any anisotropy in
   the universe.
   The remarkable fact is, that the microwave background is the same in
   every direction, to a high degree of accuracy. It wasn't until 1982,
   that differences between different directions were detected, at the
   level of one part in a thousand, with a dipole pattern on the sky.
   However, this could be interpreted as a consequence of our galaxy's
   motion through the universe, which blue shifted the microwave
   radiation in one direction, and red shifted it in the opposite
   direction. It need not represent any intrinsic anisotropy in the
   universe.
   It was not until 1992, that tiny fluctuations on angular scales of 10
   degrees, were detected by the Cosmic Background Explorer satellite,
   cobee. Since then, similar fluctuations have been found on smaller
   angular scales. The shape of the spectrum of fluctuations against
   angular scale, is still rather uncertain, but it is clear that the
   general size of the departures from uniformity, is only one part in
   ten to the five.
   This uniformity of the microwave background in different directions,
   was very difficult to understand. It seems the microwave background,
   is the last remnant of the radiation, that filled the hot early
   universe. What we observe, would have propagated freely to us, from a
   time of last scattering, when the universe was a thousandth of the
   size now. But according to the accepted hot big bang theory, the
   radiation coming from directions on the sky more than a degree apart,
   would be coming from regions of the early universe, that hadn't been
   in communication since the big bang. It was therefore truely
   remarkable, that the microwaves we observe in different directions,
   are the same to one part in ten to the five. How did different regions
   in the early universe, know to be at almost exactly the same
   temperature. It is a bit like handing in home work. If the whole class
   produce exactly the same, you can be sure they have communicated with
   each other. But according to the hot big bang model, there wasn't time
   since the big bang, for signals to get from one region to another. So
   how did all the regions, come up with the same temperature for the
   microwaves.
   If we assume that the universe is roughly homogeneous and isotropic,
   it can be described by one of the Friedmann Robertson Walker models.
   These are characterized by a scale factor S, which gives the distance
   between two neighbouring points, in the expanding universe. There are
   three kinds of Friedmann model, according to the sign of k, the
   curvature of the surfaces of constant time. If k =+1, the surfaces of
   constant time, are three spheres, and the universe is closed and
   finite in space. If k = minus 1, the surfaces have negative curvature,
   like a saddle, and the universe is infinite in spatial extent. The
   third possibility, k=0, a spatially flat universe, is of measure zero,
   but it is an important limiting case.
   Because the universe is expanding, the scale factor, S, is increasing
   with time.
   The second derivative of S, is given by the Einstein equation, in
   terms of the energy density and pressure, of matter in the universe,
   and the cosmological constant, lambda. For the moment, I will take
   lambda to be zero. For normal matter, both the energy density and
   pressure, will be positive. Thus the expansion of the universe, will
   be slowing down. In particular, for a universe dominated by radiation,
   like the early stages of the hot big bang model, the scale factor will
   go like t to the half.
   In such a model, one can ask how far one can see, before one sees
   right back to the big bang. It is easy to work out, that this is just
   the integral of one over the scale factor. For the hot big bang model,
   this integral converges. This means that a point in an early hot big
   bang universe, could have communicated only with a small region round
   it. Why then did it have almost exactly the same temperature, as
   regions far away.
   A possible explanation, was provided by the theory of inflation, which
   was put forward independently in the Soviet union, and the west,
   around 1980. The idea was to make regions able to communicate, by
   changing the expansion of the early universe, so that S double dot was
   positive, rather than negative. In other words, so that the expansion
   of the universe was being accelerated, rather than slowed down by
   gravity.
   As you can see from the Einstein equations, such accelerating
   expansion, or inflation, as it was called, required either negative
   energy, or negative pressure. One gets in a lot of trouble, if one
   allows negative energy. One would get runaway creation of particle
   pairs, one with positive energy, and the other with negative. But
   there is no reason to rule out negative pressure. That is just
   tension, which is a very common condition in the modern world.
   The original idea for inflation, was that in some way, the universe
   got trapped in what was called, a false vacuum state. A false vacuum
   state, is a Lorentz invariant meta-stable state, that has more energy
   than the true vacuum, which is taken to have zero energy density.
   Because a false vacuum is Lorentz invariant, its energy momentum
   tensor must be proportional to the metric. Since the false vacuum has
   positive energy density, the coefficient of proportionality must be
   negative. This means that the pressure in the false vacuum, is minus
   the energy density. The Einstein equations, then imply that the scale
   factor, increases exponentially with time. In such a universe, the
   integral of one over the scale factor, diverges as one goes back in
   time. This means that different regions in the early universe, could
   have communicated with each other, and come to equilibrium at a common
   state, explaining why the microwaves, look the same in different
   directions.
   The original model of inflation, which came to be known as old
   inflation, had various problems. How did the universe get into a false
   vacuum state in the first place, and how did it get out again. Various
   modifications were proposed, that went under the names of new
   inflation, or extended inflation. I won't describe them, because I
   have got into trouble in the past, about who should have credit for
   what, and because I now consider them irrelevant. As Lindeh first
   pointed out, it is not necessary for the universe to be in a false
   vacuum, to get inflation.
   A scalar field with a potential V, will have the energy momentum
   tensor shown on the screen. If the field is nearly constant in a
   region, the gradient terms will be small, and the energy momentum
   tensor, will be minus half V, times the metric. This is just what one
   needs for inflation.
   In the false vacuum case, the scalar field sits in a local minimum of
   the potential, V. In that case, the field equation allows the scalar
   field, to remain constant in space and time. If the scalar field is
   not at a local minimum, it can not remain constant in time, even if it
   is initially constant in space. However, Lindeh pointed out that if
   the potential is not too steep, the expansion of the universe, will
   slow down the rate at which the field rolls down the potential, to the
   minimum. The gradient terms in the energy momentum tensor, will remain
   small, and the scale factor will increase almost exponentially. One
   can get inflation with any reasonable potential V, even if it didn't
   have local minima, corresponding to false vacua. The work that Neil
   and I have done, is a logical extension of Andrei's idea. But I'm not
   sure if Andrei agrees with it, though I think he's coming round.
   Andrei's idea removed the need to believe that the universe began in a
   false vacuum. However, one still needed to explain, why the field
   should have been nearly constant over a region, with a value that was
   not at the minimum of the potential.
   To do this, one has to have a theory of the initial conditions of the
   universe. There are three main candidates. They are, the so called
   pre-big bang scenario, the tunneling hypothesis, and the no boundary
   proposal. In my opinion, the pre-big bang scenario is misguided, and
   without predictive power. And I feel the tunneling hypothesis, is
   either not well defined, or gives the wrong answers. But then I'm
   biased, for it was Jim Hartle and I, that were responsible for the no
   boundary proposal.
   This says that the quantum state of the universe, is defined by a
   Euclidean path integral over compact metrics, without boundary.. One
   can picture these metrics, as being like the surface of the Earth,
   with degrees of latitude, playing the role of imaginary time. One
   starts at the north pole, with the universe as a single point. As one
   goes south, the spatial size of the universe, increases like the
   lengths of the circles of latitude. The spatial size of the universe,
   reaches a maximum size at the equator, and then shrinks again to a
   point at the south pole.
   Of course, spacetime is four dimensional, not two dimensional, like
   the surface of the Earth, but the idea is much the same. I shall go
   through it in detail, because it is basic to the work I'm going to
   describe. The simplest compact four dimensional metric that might
   represent the universe, is the four sphere.
   One can give its metric in terms of coordinates, sigma, chi, theta and
   phi. One can think of sigma, as an imaginary time coordinate, and chi,
   theta and phi, as coordinates on a three sphere, that represents the
   spatial size of the universe. Again, one starts at the north pole,
   sigma =0, with a universe of zero spatial size, and expands up to a
   maximum size at the equator, sigma = pi, over 2H. But we live in a
   universe with a Lorentzian metric, like Minkowski space, not a
   Euclidean, positive definite metric.
   One therefore has to analytically continue, the Euclidean metrics used
   in the path integral, for the no boundary proposal. There are several
   ways one can analytically continue, the metric of the four sphere, to
   a Lorentzian spacetime metric. The most obvious is to follow the
   Euclidean time variable, sigma, from the north pole to the equator,
   and then go in the imaginary sigma direction, and call that real
   Lorentzian time, t. Instead of the size of the three spheres going as
   the sine of H sigma, they now go as the cosh of H t. This gives a
   closed universe, that expands exponentially with real time. At late
   times, the expansion will change from being exponential, to being
   slowed down by matter in the normal way. This departure of the scale
   factor from a cosh behavior, will occur because the original Euclidean
   four sphere, was not perfectly round. But the universe would still be
   closed, however deformed the four sphere.
   For nearly 15 years, I believed that the no boundary proposal,
   predicted that the universe was spatially closed. I also believed that
   the cosmological constant was zero, because it seemed unreasonable to
   suppose that it was less than the observational limit of 10 to the
   minus 120 Planck units, unless it were exactly zero. But the Einstein
   equations, relate the energy density in the universe, plus lambda, to
   the rate of expansion, and the curvature, k, of the surfaces of
   constant time.
   Define omega matter and omega lambda, to be the density and lambda,
   divided by the critical value. If the universe is closed, that is,
   k=+1, omega matter plus omega lambda, must be greater than one.
   Observations of luminous matter, like stars and gas clouds give an
   omega matter of about 0 point 0 2. We know that galaxies and clusters
   of galaxies, must contain non luminous, or dark matter, but the best
   estimate of this, is that it contributes at most 0 point 2 of the
   critical density. Still, Eddington once said, if your theory doesn't
   agree with the observations, don't worry. The observations are
   probably wrong. But if your theory doesn't agree with the second law
   of thermodynamics, forget it. I firmly believed in the no boundary
   proposal, and I thought it implied that the universe had to be closed.
   Since a closed universe, is not incompatible with the second law of
   thermodynamics, I was sure the observers had missed something, and
   there really was enough matter to close the universe. At that time, I
   didn't take the seriously the possibility of a small cosmological
   constant.
   The observations do not yet indicate that the universe is definitely
   open, or that lambda is non zero, but it is begining to look like one
   or the other, if not both. I won't go through all the observations,
   but shall just show what I consider to be the most significant pieces
   of evidence.
   The first is the distribution of large scale inhomogeneities in the
   universe. On the very largest scales, this can be measured by
   fluctuations in the microwave background, and on smaller scales by the
   galaxy galaxy correlation function. One can then try and fit these
   observations, with the predictions of inflationary theory. If one
   assumes the universe is filled with cold dark matter, the predicted
   spectrum of irregularities, depends on a quantity gamma. This is the
   product of omega matter, with the Hubble constant, or rate of
   expansion, in units of a hundred kilometers per second, per
   Megaparsec. (Astronomers use funny units). It is generally believed
   that the Hubble constant, is somewhere between 50 and 100 of those
   funny units. Thus if omega matter is one, gamma must be at least 0
   point 5. As you can see, a gamma of 0 point 5, would predict much less
   irregularity on large angles, than is observed. One can get a
   reasonable fit to the observations, with a gamma of 0 point 2. If
   omega matter were one, this would imply a Hubble constant of only 20.
   As a theorist, I would be happy with such a figure, because it would
   make the universe older, and remove a possible conflict with the ages
   of some stars. But the observers claim the Hubble constant, has to be
   in the range, 50 to 100. This would imply that omega matter is at most
   0 point 4.
   Thus dynamical measurements, give us a vertical strip in the omega
   matter, omega lambda plane.
   One can obtain further limits in this plane, from observations of
   supernova. Type 1 supernova, are standard candles. That is, the total
   energy in the explosion, is always the same, within a factor close to
   one. One can thus use their observed brightness, as a distance
   measurement, and compare it with their red shifts. This gives the
   limits shown on the diagram, for which I'm grateful to Ned Wright and
   Shawn Carol. The yellow, red and green areas represent the formal
   errors, and the large pink area, other possible errors. Also shown in
   blue, are the limits set by the position of a peak in the angular
   spectrum, of the variations of the microwave background. As you can
   see, the observations suggest that the universe is close to the open
   closed divide, but with a non zero lambda.
   Despite these indications of a low density lambda universe, I
   continued to believe that the cosmological constant was zero, and the
   no boundary proposal, implied that the universe must be closed. Then
   in conversations with Neil Turok, I realized there was another way of
   looking at the no boundary universe, that made it appear open. One
   starts with the point that Andrei Lindeh made, that inflation doesn't
   need a false vacuum, a local minimum of the potential. But if the
   scalar field is not at a stationary point of the potential, then it
   can not be constant on an instanton, a Euclidean solution of the field
   equations. In turn, this implies that the instanton can't be a
   perfectly round four sphere. A perfectly round four sphere, would have
   the symmetry group, O5.
   But with a non constant scalar field, the largest symmetry group that
   an instanton can have, is O4. In other words, the instanton is a
   deformed four sphere.
   One can write the metric of an O4 instanton, in terms of a function, b
   of sigma. Here b is the radius of a three sphere of constant distance,
   sigma, from the north pole of the instanton. If the instanton were a
   perfectly round four sphere, b would be a sine function of sigma. It
   would have one zero at the north pole, and a second at the south pole,
   which would also be a regular point of the geometry. However, if the
   scalar field at the north pole, is not at a stationary point of the
   potential, it will be almost constant over most of the four sphere,
   but will diverge near the south pole. This behavior is independent of
   the precise shape of the potential. The non constant scalar field,
   will cause the instanton not to be a perfectly round four sphere, and
   in fact there will be a singularity at the south pole. But it will be
   a very mild singularity, and the Euclidean action of the instanton
   will be finite.
   This Euclidean instanton, has been described as the universe begining
   as a pea. In fact, a pea is quite a good image for a deformed sphere.
   Its size of a few thousand Planck lengths, makes it a very petty pea.
   But the mass of the matter it contains, is about half a gram, which is
   about right for a pea.
   I actually discovered this pea instanton in 1983, but I thought it
   could describe the birth of close universes only.
   To get a closed universe, one starts with sigma =0 at the north pole,
   and proceeds to the equator, or rather the value of sigma at which the
   radius, b, of the three sphere is maximum. One then analytically
   continues sigma in the imaginary direction, as Lorentzian time. As I
   described earlier, this gives a closed universe with a scale factor
   that initially goes like cosh t. The scalar field, will have a small
   imaginary part, but that can be corrected by giving the initial value
   of the scalar field at the north pole, a small imaginary part.
   According to the no boundary proposal, the relative probability of
   such a closed universe, is e to minus twice the action of the part of
   the pea instanton, between the north pole, and the equator. Notice
   that as this part, doesn't contain the singularity at the south pole,
   there is no ambiguity about the action of a singular metric. The
   action of this part of the instanton, is negative, and is more
   negative, the bigger the pea. Thus the probability of the pea, is
   bigger, the bigger the pea. The negative sign of the action, may look
   counter intuitive, but it leads to physically reasonable consequences.
   As I said, I thought the no boundary proposal, implied that the
   universe had to be spatially closed, and finite in size. But Neil
   Turok and I, realized his ideas on open inflation, could be fitted in
   with the no boundary proposal. The universe would still be closed and
   finite, in one way of looking at it. But in another, it would appear
   open and infinite.
   Let's go back to the metric for the pea instanton, and analytically
   continue it in a different way. As before, one analytically continues
   the Euclidean latitude coordinate, in the imaginary direction, to
   become a Lorentzian time, t. The difference is that one goes in the
   imaginary sigma direction at the north pole, rather than the equator.
   One also continues the coordinate, chi, in the imaginary direction, as
   a coordinate, psi. This changes the three sphere, into a hyperbolic
   space. One therefore gets an exponentially expanding open universe.
   One can think of this open universe, as a bubble in a closed, de
   Sitter like universe. In this way, it is similar to the single bubble
   inflationary universes, that have been proposed by a number of
   authors. The difference is, the previous models all required carefully
   adjusted potentials, with false vacuum local minima. But the pea
   instanton, will work for any reasonable potential. The price one pays
   for a general potential, is a singularity at the south pole. In the
   analytically continued Lorentzian spacetime, this singularity would be
   time like, and naked. One might think that this naked singularity,
   would mean one couldn't evaluate the action of the instanton, or of
   perturbations about it. This would mean that one couldn't predict the
   quantum fluctuations, or what would happen in the universe. However,
   the singularity at the south pole, the stalk of the pea, is so mild,
   that the actions of the instanton, and of perturbations around it, are
   well defined. This means one can determine the relative probabilities
   of the instanton, and of perturbations around it. The action of the
   instanton itself, is negative, but the effect of perturbations around
   the instanton, is to increase the action, that is, to make the action
   less negative. According to the no boundary proposal, the probability
   of a field configuration, is e to minus its action. Thus perturbations
   around the instanton, have a lower probability, than the unperturbed
   background. This means that quantum fluctuation are suppressed, the
   bigger the fluctuation, as one would hope. On the other hand,
   according to the tunneling hypothesis, favored by Vilenkin and Lindeh
   , probabilities are proportional to e to the ~plus action. This would
   mean that quantum fluctuation would be ~enhanced, the bigger the
   fluctuation. There is no way this could lead to a sensible description
   of the universe. Lindeh therefore proposes to take e to the ~plus
   action, for the probability of the background universe, but e to the
   ~minus action, for the perturbations. However, there is no invariant
   way, in which one can divide the action, into a background part, and a
   part due to fluctuations. So Lindeh's proposal, does not seem well
   defined in general. By contrast, the no boundary proposal, is well
   defined. Its predictions may be surprising, but they are not obviously
   wrong.
   To recapitulate. A general potential, without false vacuums, or local
   minimums, leads to the pea instanton. This can be analytically
   continued, to either an open, or a closed universe. The no boundary
   proposal, then allows one to calculate the relative probabilities of
   different backgrounds, and the quantum fluctuations about them.
   There isn't just a single pea instanton, but a whole family of them,
   labeled by different values of the scalar field at the north pole. The
   higher the value of the potential at the north pole, the smaller the
   instanton, and the less negative the value of the action. Thus the no
   boundary proposal, predicts that large instantons, are more probable
   than small ones. This is a problem, because large instantons, will
   lead to a shorter period of exponential expansion or inflation, than
   small ones. In the closed universe case, a short period of inflation,
   would mean the universe would recollapse before it reached the present
   size and density. On the other hand, an open universe with a short
   period of inflation, would become almost empty early on.
   Clearly, the universe we live in, didn't collapse early on, or become
   almost empty. So we have to take account of the anthropic principle,
   that if the universe hadn't been suitable for our existence, we
   wouldn't be asking why it is, the way it is. Many physicists don't
   like the anthropic principle, but I think some version of it is
   essential, in quantum cosmology. M theory, or whatever the ultimate
   theory is, seems to allow a very large number of possible solutions,
   and compactifications. One has to have some criterion, for discarding
   most of them. Otherwise, why isn't the universe, eleven dimensional
   Minkowski space.
   The approach Neil Turok and I took, was to invoke the weakest version
   of the anthropic principle. We adopted Bayes statistics. In this, one
   starts with an a-priori probability distribution, and then modifies it
   in light of ones knowledge of the system. In this case, we took the
   a-priori distribution, to be the e to the minus action, predicted by
   the no boundary proposal. We then modified it, by the probability that
   the model contained galaxies, which are presumably a necessary
   condition, for the existence of intelligent observers. An open
   universe, has an infinite spatial volume. Thus the total number of
   galaxies in an open universe, would always be infinite, no matter how
   low the probability of finding a galaxy, in a given comoving volume.
   One therefore can not weight the a-priori probability, given by the no
   boundary proposal, by the total number of galaxies in the universe.
   Instead, we weighted by the comoving density of galaxies, predicted
   from the growth of quantum fluctuations, about the pea instanton. This
   gives a modified probability distribution for omega, the present
   density, divided by the critical density. For the open models, this
   probability distribution, is sharply peaked at an omega of about zero
   point zero one. This is lower than is compatible with the
   observations, but it is not such a bad miss. As far as I'm aware, this
   is the first attempt to ~predict a value of omega for an open
   universe, rather than fine tune a false vacuum potential, to obtain a
   value in the range indicated by observation.
   The anthropic arguments we have used, are fairly crude, and could be
   refined. But the best hope of getting a more realistic omega, seems to
   be to include other fields. Eleven dimensional super gravity, which is
   the best candidate we have for a theory of everything, has a three
   form potential, with a four form field strength. When dimensionally
   reduced to four dimensions, this can act as a cosmological constant.
   For a real four form in dimensions, the contribution to the
   cosmological constant is negative. It can therefore cancel the
   positive contribution to the cosmological constant, that must arise
   because super symmetry is broken, in the universe we live in. Indeed,
   super symmetry breaking, is a necessary condition for life. But
   galaxies will not form, unless the total cosmological constant, is
   almost zero. Thus the anthropic principle fixes the value of the four
   form field strength, which is a free parameter of the theory, so it
   almost cancels the positive contribution from symmetry breaking. But
   it need not cancel it exactly. The anthropic requirement, can probably
   be satisfied by any omega lambda between about minus one point five,
   and plus one point five, with a fairly flat probability distribution.
   This is consistent with the observations. My student, Harvey Reall and
   I, are now working on an eleven dimensional supergravity version of
   the pea instanton. One gets a reduced action in four dimensions with a
   four form, and two scalar fields, which describe the size, and the
   squashing of a seven sphere.
   The squashing scalar field, phi, has a potential with a minimum at the
   round metric, and a maximum at the squashed sphere with an Einstein
   metric. One can get a pea instanton, by starting phi on the
   exponential wall on the right. This would produce an inflationary
   universe, in which the squashing ran down to the round seven sphere.
   The scalar field that represents the size of the seven sphere, has a
   potential that looks unstable. However, if one takes into account the
   back reaction of the scalar field on the four form, the effective
   potential becomes stable. This looks good, but the potentials are too
   steep to give enough inflation. Maybe if we can include the dynamical
   effects of symmetry breaking, we can get something more reasonable.
   The aim is to find a description of the origin of the universe, on the
   basis of fundamental theory.
   Assuming that one can find a model that predicts a reasonable omega,
   how can we test it by observation. The best way is by observing the
   spectrum of fluctuations, in the microwave background. This is a very
   clean measurement of the quantum fluctuations, about the initial
   instanton. However, there is an important difference between our
   instanton, and previous proposals for open inflation. They have all
   assumed false vacuum potentials, and have used the Coleman De Lucia
   instanton, which is non singular. However, our instanton has a
   singularity at the south pole. There has been a lot of discussion of
   this singularity. In the Lorentzian analytically continued spacetime,
   the singularity is time like and naked. People have worried about this
   singularity, because it seemed to make the spacetime non predictable.
   Anything could come out of the singularity. However, perturbations of
   the Euclidean instanton, have finite action if and only, they obey a
   Dirichelet boundary condition at the singularity. Perturbation modes
   that don't obey this boundary condition, will have infinite action,
   and will be suppressed. Support for this boundary condition, has come
   from the work of Garriga, who has shown that in some cases at least,
   the singularity in the instanton, is just an artefact of Kaluza Klein
   reduction from higher dimensional spacetimes. In these situations,
   perturbations would obey this Dirichelet boundary condition.
   When one analytically continues to Lorentzian spacetime, the
   Dirichelet boundary condition, implies that perturbations reflect at
   the time like singularity. This has a significant effect on the two
   point correlation function of the perturbations. I show preliminary
   calculations that Neil and a student have made, for the case of omega
   equals zero point three.
   The first shows the two point correlation function of the microwave
   background, as a function of angle, for our instanton, and for false
   vacuum open inflation. The difference, which is plotted on a magnified
   scale, is like a step function at 30 degrees, the angle subtended by
   the curvature radius, on the surface of last scattering. The next
   graph, shows the power spectrum of this correlation function. You see
   it has small oscillations, that come from the Fourier transform, of
   the step function.
   The present observations of the microwave fluctuations, are not
   sensitive enough to detect this effect. But it may be possible with
   the new observations that will be coming in, from the map satellite in
   two thousand and one, and the Planck satellite in two thousand and
   six. Thus the no boundary proposal, and the pea instanton, are real
   science. They can be falsified by observation.
   I will finish on that note.